Make a sketch of the block-diagonalized portion of S¹ F S for the E-mode block if S is based on the symmetry functions of Exercise 1.8. Explain why the two 2 2 blocks must have the same eigenvalues.

Exercise 1.8

The decomposition of (Exercise 1.4) contains the E IR three times. Therefore there are three symmetry functions for each row. One of the row1 symmetry functions is the translation function Tx . The two remaining row-1 symmetry functions correspond to vibrations. (1) The displacements ( -functions) of these two row-1 functions must leave the center-of-mass stationary, and (2) the corresponding -functions must be orthogonal to all of the other symmetry functions.

(a) Show that the row-1 and row-2 -functions for the E vibrations satisfying conditions (1) and (2) are

(b) For the , make a sketch of the components of each of the six -functions and give the relative amplitudes of the displacements.

(c) Make a sketch of the sum of and and the sum of and .